Computer implemented system for determining the most profitable distribution policy for a single period inventory system subject to a stochastic metric constraint, optimization application therefor, and method therefor, and decision support tool for facilitating user determination of a distribution policy for a single period inventory system

ABSTRACT

Computer implemented system for determining the most profitable distribution policy for a single period inventory system subject to a stochastic metric constraint, optimization application therefor, and method therefor, and including a Decision Support Tool for facilitating user determination of a distribution policy for a single period inventory system.

FIELD OF THE INVENTION

[0001] The invention is in the field of determining distributionpolicies for single period inventory systems.

GLOSSARY OF TERMS

[0002] The following alphabetically listed terms together with theiracronyms are employed in the description and claims of this applicationwith respect to the present invention:

[0003] Availability A_(ij), Optimal Availability A_(ij)*, and OptimalAvailability Matrix A*

[0004] Availability A_(ij) is an industry term referring to theprobability of completely satisfying the demand for an i^(th) consumeritem where i=1, 2, . . . , m at a j^(th) location where j=1, 2, . . . ,n of a single period inventory system without an occurrence of a selloutdue to insufficient draw at that location. In mathematical terms,A_(ij)=F(λ_(ij),D_(ij)) where F is the cumulative probabilitydistribution function (cdf) of demand for the i^(th) consumer item atthe j^(th) location, and λ_(ij) and D_(ij) are its mean demand and draw,respectively. The probability distribution function in the discrete caseand the probability density function in the continuous case are bothdenoted by the letter “f”. The optimal availability A_(ij)* of an i^(th)consumer item at a j^(th) location of a single period inventory systemis a function of its unit retail price p_(ij), its unit production costc_(ij), its unit return cost g_(ij), and its unit stockout cost b_(ij)and is that availability which maximizes the expected total profitrealizable for that i^(th) j^(th) item-location of the single periodinventory system. A unit return cost g_(ij) is the difference betweenthe expenses incurred upon returning a unit and its salvage value, andconsequently it may assume either a positive or negative value. Anoptimal availability matrix A* is the matrix of optimal availabilitiesA_(ij)* for a single period inventory system, and degenerates to asingle so-called optimal common availability A* in the case that therespective values of p_(ij), c_(ij), g_(ij) and b_(ij) are the same forall its i^(th) j^(th) item-locations.

[0005] Demand X_(ij), Mean Demand λ_(ij), and Mean Demand Matrix λ

[0006] The demand process for a consumer item at a location has a randombut non-stationary nature, and therefore cannot be subjected to ensembleinferences based on a single realization. Mean demands λ_(ij) for aconsumer item at a location over time are presupposed to be the outcomeof a stochastic process which can be simulated by a forecast modelwhilst the demand X_(ij) for an i^(th) consumer item at a j^(th)location of a single period inventory system at a future point in timeis a random variable with a conditional probability distributionconditioned on its mean demand λ_(ij) at that point in time. A meandemand matrix λ is a matrix of mean demands λ_(ij).

[0007] Distribution Policy

[0008] A distribution policy is the delivered quantities of each i^(th)consumer item where i=1, 2, . . . , m at each j^(th) location where j=1,2, . . . , n of a single period inventory system in accordance with apredetermined business strategy. The most profitable distribution policyfor a single period inventory system is realized by an optimal drawmatrix D*. The most profitable distribution policy for a single periodinventory system subject to a constraint is realized by an optimalconstrained draw matrix D{circumflex over ( )}.

[0009] Draw D_(ij), Draw Matrix D, Optimal Draw Matrix D*, and WeightedTotal Draw TD

[0010] Draw D_(ij) is an industry term referring to the deliveredquantity of an i^(th) consumer item where i=1, 2, . . . , m at a j^(th)location where j=1, 2, . . . , n of a single period inventory system. Adraw matrix D is the matrix of draws D_(ij) for all i^(th) j^(th)item-locations of a single period inventory system. The optimal drawmatrix D* for a single period inventory system is the draw matrixmaximizing the expected total profit realizable by a distribution policytherefor. The weighted total draw TD of all m consumer items at all nlocations of a single period inventory system is given by ΣΣw_(ij)D_(ij)where w_(ij) are the weights correspondingly associated with its i^(th)j^(th) item-locations.

[0011] Returns R(λ_(ij),D_(ij)), and Expected Weighted Total ReturnsER(λ,D)

[0012] Returns R(λ_(ij),D_(ij)) is an industry term referring to thenumber of unsold units of an i^(th) consumer item at a j^(th) locationof a single period inventory system, and is given byR(λ_(ij),D_(ij))=max(D_(ij)−X_(ij), 0) where D_(ij), X_(ij), and λ_(ij)are its draw, demand, and mean demand, respectively, at that location.The expected weighted total returns ER(λ,D) of all m consumer items atall n locations of a single period inventory system is given byER(λ,D)=ΣΣw_(ij)ER(λ_(ij),D_(ij)) where w_(ij) are the weightscorrespondingly associated with its it j^(th) item-locations, andER(λ_(ij),D_(ij)) is the expected value of R(λ_(ij),D_(ij)). For aPoisson distribution of demand,ER(λ,D)=ΣΣw_(ij)[D_(ij)f(λ_(ij),D_(ij)−1)+(D_(ij)−λ_(ij))F(λ_(ij),D_(ij)−2)].

[0013] Sales S(λ_(ij),D_(ij)) and Expected Weighted Total Sales ES(λ,D)

[0014] Sales S(λ_(ij),D_(ij)) refers to the quantity of sold items of ani^(th) consumer item at j^(th) location of a single period inventorysystem as upper bounded by the draw D_(ij) at that location at eachpoint in time, and is given byS(λ_(ij),D_(ij))=min(D_(ij),X_(ij))=D_(ij)−R(λ_(ij),D_(ij)), whereD_(ij), X_(ij), and λ_(ij) are its draw, demand, and mean demand,respectively, at that location. The expected weighted total salesES(λ,D) of all m consumer items at all n locations of a single periodinventory system is given by ES(λ,D)=ΣΣw_(ij)ES(λ_(ij),D_(ij)) wherew_(ij) are the weights correspondingly associated with its i^(th) j^(th)item-locations, and ES(λ_(ij),D_(ij)) is the expected value ofS(λ_(ij),D_(ij)).

[0015] Single Period Inventory Systems

[0016] Single period inventory systems are largely concerned withconsumer items having a limited shelf life at the end of which an itemloses most, if not all, of its consumer value, and the stock of which ata j^(th) location is not replenished to prevent an occurrence of asellout. Such consumer items can include perishable goods, for example,fruit, vegetables, flowers, and the like, and fixed lifetime goods, forexample, printed media publications, namely, daily newspapers, weeklies,monthlies, and the like. Two common degenerate problems of single periodinventory systems are known in the industry as the so-called“newsvendor” problem i.e. the sale of the same item throughout amulti-location single period inventory system and the so-called“knapsack” problem i.e. the sale of different items at the samelocation.

[0017] Stockout ST(λ_(ij),D_(ij)), and Expected Weighted Total StockoutEST(λ,D)

[0018] Stockout ST(λ_(ij),D_(ij)) is the quantity of unsatisfied demandfor an i^(th) consumer item at a j^(th) location of a single periodinventory system, and is given by ST(λ_(ij),D_(ij))=max(X_(ij)−D_(ij),0)=X_(ij)−S(λ_(ij),D_(ij)) where D_(ij), X_(ij), and λ_(ij) are itsdraw, demand, and mean demand, respectively, at that location. Theexpected weighted total stockout EST(λ,D) of all m consumer items at alln locations of a single period inventory system is given byEST(λ,D)=ΣΣw_(ij) EST(λ_(ij),D_(ij))=ΣΣw_(ij)(λ_(ij)−ES(λ_(ij),D_(ij))), where w_(ij) are the weightscorrespondingly associated with its i^(th) j^(th) item-locations, andEST(λ_(ij),D_(ij)) is the expected value of ST(λ_(ij),D_(ij)).

[0019] Weights w_(ij), and Weighted Totals

[0020] Weights w_(ij) are employed in the calculation of weighted totalsto differentiate between item-locations in terms of their relativeimportance to satisfy some business objectives such as cost, goodwill,exposure to preferred populations, and the like. Thus, for example, theweighted total draw ΣΣw_(ij)D_(ij) can represent a total budget in thecase that w_(ij)'s haved pecuniary values.

BACKGROUND OF THE INVENTION

[0021] One computer implemented approach for calculating a demandforecast involves defining a so-called demand forecast tree capable ofbeing graphically represented by a single top level node with at leasttwo branches directly emanating therefrom, each branch having at leastone further node. The demand forecast is computed on the basis ofhistorical sales data typically associated with bottom level nodes of ademand forecast tree by a forecast engine capable of determining amathematical simulation model for a demand process. One such forecastengine employing statistical seasonal causal time series models of countdata is commercially available from Demantra Ltd, Israel, under the nameDemantra™ Demand Planner.

[0022] Demand forecast applications include determining the optimal drawmatrix D* to maximize the expected total profit (ETP) realizable by adistribution policy for a single period inventory system given by:$\begin{matrix}{{ETP} = {{\sum\limits_{i\quad j}{{Ep}\left( D_{i\quad j} \right)}} = {\sum\limits_{i\quad j}{\quad\left\lbrack {{\left( {p_{i\quad j} - c_{i\quad j}} \right)D_{i\quad j}} - {\left( {p_{i\quad j} - g_{i\quad j}} \right){{ER}\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}} - {b_{i\quad j}{{EST}\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}}} \right\rbrack}}}} & {{Eqn}.\quad (1)}\end{matrix}$

[0023] where p_(ij) is the unit retail price of an i^(th) consumer itemat a j^(th) location of the single period inventory system, c_(ij) isits unit production cost, g_(ij) is its unit return cost when unsold,and b_(ij) is its unit stockout cost. Derived from Eqn. (1), the optimaldraw matrix D* for a single period inventory system is calculated usingoptimal availabilities A_(ij)* where: $\begin{matrix}{A_{i\quad j}^{*} = {{F\left( {\lambda_{i\quad j},D_{i\quad j}^{*}} \right)} = {\frac{p_{i\quad j} - c_{i\quad j} + b_{i\quad j}}{p_{i\quad j} - g_{i\quad j} + b_{i\quad j}}.}}} & {{Eqn}.\quad (2)}\end{matrix}$

[0024] In the case of the above-mentioned “newsvendor” and the“knapsack” problems, Eqn. (1) respectively degenerates to:$\begin{matrix}{{ETP} = {{\sum\limits_{j}{EP}_{j}} = {\sum\limits_{j}\left\lbrack {{\left( {p_{j} - c_{j}} \right)D_{j}} - {\left( {p_{j} - g_{j}} \right){{ER}\left( {\lambda_{j},D_{j}} \right)}} - {b_{j}{{EST}\left( {\lambda_{j},D_{j}} \right)}}} \right\rbrack}}} \\{\quad {and}} \\{{ETP} = {{\sum\limits_{i}{EP}_{i}} = {\sum\limits_{i}{\left\lbrack {{\left( {p_{i} - c_{i}} \right)D_{i}} - {\left( {p_{i} - g_{i}} \right){{ER}\left( {\lambda_{i\quad},D_{\quad i}} \right)}} - {b_{i}{{EST}\left( {\lambda_{i},D_{i}} \right)}}} \right\rbrack.}}}}\end{matrix}$

[0025] A distribution policy for a single period inventory system isoften subject to one or more deterministic metric constraints, forexample, a maximum total draw, a maximum budget, and the like, whichnecessitate an optimal constrained draw matrix denoted D{circumflex over( )} whose total draw is typically less than the total draw of theoptimal draw matrix D*. Two common approaches for solving such types ofproblems in the field of single-period inventory systems are theLagrange multiplier approach as described in Silver, E., D. Pyke, and R.Peterson: Inventory Management and Production Planning and Scheduling,3d ed., Wiley, NY, 1998, pgs. 406-422, and the one-by-one allocation orremoval of draw units as discussed in Hadley, G., and T. M. Whitin,Analysis of Inventory Systems, Prentice-Hall, 1963, pgs. 304-307, thecontents of which are incorporated herein by reference.

SUMMARY OF THE INVENTION

[0026] Broadly speaking, the present invention provides a novel computerimplemented system for determining the most profitable distributionpolicy for a single period inventory system subject to a stochasticmetric constraint which reflects a marketing objective as opposed to ahitherto deterministic metric constraint which reflects a production ora budgetary objective, optimization application therefor, and methodtherefor. The computer implemented system may also include a computerimplemented Decision Support Tool for facilitating user determination ofa distribution policy for a single period inventory system.

[0027] A first preferred stochastic metric constraint in accordance withthe present invention is a minimum threshold imposed on the expectedweighted total sales ES(λ,D) (hereinafter referred to “sales target”) ofa distribution policy for a single period inventory system. The optimalconstrained draw matrix D{circumflex over ( )} for the single periodinventory system is required to satisfy the condition ES(λ,D{circumflexover ( )})≧G for some pre-determined threshold G. In the case that theoptimal draw D* does not satisfy the sales target G, namely, ES(λ,D*)<G,the total draw of the optimal constrained draw matrix D{circumflex over( )} is greater than the total draw of the optimal draw matrix D*.

[0028] A second preferred stochastic metric constraint in accordancewith the present invention is a maximum threshold imposed on theexpected weighted total stockouts EST(λ,D) (hereinafter referred to“stockout constraint”) of a distribution policy for a single periodinventory system. The optimal constrained draw matrix D{circumflex over( )} for the single period inventory system is required to satisfy thecondition EST(λ,D {circumflex over ( )})≦H for some pre-determinedthreshold H. In the case that the optimal draw D* does not satisfy thestockout constraint H, namely, that EST(λ,D*)>H, the total draw of theoptimal constrained draw matrix D{circumflex over ( )} is also greaterthan the total draw of the optimal draw matrix D*.

[0029] A third preferred stochastic metric constraint in accordance withthe present invention is a maximum threshold imposed on the expectedweighted total returns ER(λ,D) (hereinafter referred to as “returnsconstraint”) for example, to comply with compulsory regulations, of adistribution policy for a single period inventory system. The optimalconstrained draw matrix D{circumflex over ( )} for the single periodinventory system is required to satisfy the condition ER(λ,D{circumflexover ( )})≦V for some pre-determined threshold V. In the case that theoptimal draw D* does not satisfy the returns constraint V, namely,ER(λ,D*)>V, the total draw of the optimal constrained draw matrixD{circumflex over ( )} is less than the total draw of the optimal drawmatrix D*.

[0030] The optimal constrained draw matrix D{circumflex over ( )} for asingle period inventory system subject to any of the above threestochastic metric constraints may be iteratively arrived at using theLagrange multiplier approach. The Lagrange systems of equations to benumerically solved for determining the most profitable distributionpolicy for the three different stochastic metric constraints are derivedhereinbelow from Eqn. (1) using the following notation: In the case ofthe sales target G, L(D_(ij))=Σw_(ij)(D_(ij)−ER(λ_(ij),D_(ij)))−G, inthe case of the stockout constraint H,L(D_(ij))=Σw_(ij)EST(λ_(ij),D_(ij))−H, and in the case of the returnsconstraint V, L(D_(ij))=Σw_(ij)ER(λ_(ij),D_(ij))−V.

[0031] Introducing the Lagrange multiplier M, the following functionshave to be differentiated with respect to D_(ij) and M:

[0032] Z(D_(ij), M)=EP (D_(ij))+M·L(D_(ij)) for each ij Noting that${{\frac{\partial_{i\quad j}\quad}{\partial D_{i\quad j}}{{ER}\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}} = {{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}\quad {and}}}\quad$${{\frac{\partial_{i\quad j}\quad}{\partial D_{i\quad j}}{{EST}\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}} = {{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}\quad - 1}}\quad$

[0033] where F is the cdf of demand, then: $\begin{matrix}{{\frac{\partial_{i\quad j}\quad}{\partial D_{i\quad j}}{{EP}\left( D_{i\quad j} \right)}} = {p_{i\quad j} - c_{i\quad j} - {\left( {p_{i\quad j} - g_{i\quad j}} \right){F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}} - {b_{i\quad j}\left( {{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)} - 1} \right)}}} \\{{= {p_{i\quad j} - c_{i\quad j} + b_{i\quad j} - {\left( {p_{i\quad j} - g_{i\quad j} + b_{i\quad j}} \right){F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}\quad {for}\quad {each}\quad i}}},j}\end{matrix}$

[0034] The second term in Z differentiates with respect to D_(ij) in thecase of the sales target G to Mw_(ij)(1−F(λ_(ij),D_(ij))), in the caseof the stockout constraint H to Mw_(ij)(F(λ_(ij),D_(ij))−1), and in thecase of the returns constraint V to Mw_(ij)(F(λ_(ij),D_(ij)). Thederivatives in Z with respect to M equal L(D_(ij)) in all three cases.Equating the derivative to 0 in each case, in the case of the salestarget G,${{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)} = {\frac{p_{i\quad j} - c_{i\quad j} + b_{i\quad j} + {Mw}_{i\quad j}}{p_{i\quad j} - g_{i\quad j} + b_{i\quad j} + {Mw}_{i\quad j}}\quad {for}\quad {all}\quad i}},j$

[0035] whilst in the case of the stockout constraint H,${{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)} = {\frac{p_{i\quad j} - c_{i\quad j} + b_{i\quad j} - {Mw}_{i\quad j}}{p_{i\quad j} - g_{i\quad j} + b_{i\quad j} - {Mw}_{i\quad j}}\quad {for}\quad {all}\quad i}},j$

[0036] whilst in the case of the returns constraint V,${{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)} = {\frac{p_{i\quad j} - c_{i\quad j} + b_{i\quad j}}{p_{i\quad j} - g_{i\quad j} + b_{i\quad j} - {Mw}_{i\quad j}}\quad {for}\quad {all}\quad i}},j$

[0037] To determine the optimal constrained draw matrix D{circumflexover ( )}, these equations have to be solved along with their respectiveLagrange adjoints L(D_(ij))=0 as follows:

Σw _(ij)(_(D) _(ij−ER)(λ_(ij) ,D _(ij)))=G  (1)

Σw _(ij) EST(λ _(ij) ,D _(ij))=H  (2)

Σw _(ij) ER(λ _(ij) ,D _(ij))=V  (3)

[0038] Alternatively, the optimal constrained draw matrix D{circumflexover ( )} in the case of either the sales target G or the stockoutconstraint H may be arrived by the one-by-one allocation of additionaldraw units to the draws D_(ij)* of i^(th) j^(th) item-locations of asingle period inventory system as determined by the most profitabledistribution policy therefor in the event that D* does not satisfy thesales target or the stockout constraint respectively Each additionaldraw unit is allocated so as to result in the least decrease in theexpected total profit attributable thereto. Conversely, the optimalconstrained draw matrix D{circumflex over ( )} in the case of thereturns constraint V may be arrived by the one-by-one removal of drawunits from the draws D_(ij)* of i^(th) j^(th) item-locations of a singleperiod inventory system as determined by the most profitabledistribution policy therefor in the event that D* does not satisfy thereturns constraint. Each removed draw unit is also removed so as toresult in the least decrease in the expected total profit attributablethereto.

BRIEF DESCRIPTION OF THE DRAWINGS

[0039] In order to better understand the invention and to see how it canbe carried out in practice, preferred embodiments will now be described,by way of non-limiting examples only, with reference to the accompanyingdrawings in which:

[0040]FIG. 1 is a pictorial representation showing a demand forecasttree for calculating demand forecast information for five differentconsumer items;

[0041]FIG. 2 is a table showing historical sales data associated withthe demand forecast tree of FIG. 1;

[0042]FIG. 3 is a block diagram of a computer implemented system fordetermining the most profitable distribution policy for a single periodinventory system subject to a stochastic metric constraint, andincluding a Decision Support Tool for facilitating user determination ofa distribution policy for a single period inventory system;

[0043]FIG. 4 is a flow chart for determining the most profitabledistribution policy to meet a sales target G for a single item,multi-location single period inventory system in accordance with theLagrange multiplier approach;

[0044]FIG. 5 is a flow chart for determining the most profitabledistribution policy to meet a sales target G for a single item,multi-location single period inventory system in accordance with theone-by-one allocation of additional draw units approach;

[0045]FIG. 6 is a flow chart for determining the most profitabledistribution policy to meet a stockout constraint H for a single item,multi-location single period inventory system in accordance with theLagrange multiplier approach;

[0046]FIG. 7 is a flow chart for determining the most profitabledistribution policy to meet a stockout constraint H for a single item,multi-location single period inventory system in accordance with theone-by-one allocation of additional draw units approach;

[0047]FIG. 8 is a flow chart for determining the most profitabledistribution policy to meet a returns constraint V for a single item,multi-location single period inventory system in accordance with theLagrange multiplier approach;

[0048]FIG. 9 is a flow chart for determining the most profitabledistribution policy to meet a returns constraint V for a single item,multi-location single period inventory system in accordance with theone-by-one removal of draw units approach;

[0049]FIG. 10 is a table summarizing the results of the iterationsaccording to the Lagrange multiplier approach for meeting a sales targetG-72 newspaper copies in the event of a mean demandvector),=λ(10,21,42);

[0050]FIG. 11 is a table summarizing the results of the iterationsaccording to the one-by-one allocation of additional draw units approachfor meeting a sales target G=72 newspaper copies in the event of a meandemand vector =λ(10,21,42); and

[0051]FIG. 12 is a graph showing plots of the Decision Support Tool ofthe present invention for facilitating user determination of adistribution policy for a single item, multi-location single periodinventory system.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0052]FIG. 1 shows an exemplary demand forecast tree 1 having a singletop level node (00) with five branches A, B, C, D and E forcorrespondingly representing the sale of Item I (top level-1 node (10))at Locations 1 and 2 (bottom level nodes (11) and (21)), Item II (toplevel-1 node (20)) at Locations 1 and 3 (bottom level nodes (21) and(23)), Item III (top level-1 node (30)) at Location 1, 2 and 3 (bottomlevel nodes (31), (32) and (33)), Item IV (top level-1 node (40)) alsoat Locations 1, 2 and 3 (bottom level nodes (41), (42) and (43)); andItem V (top level-1 node (50)) at Location 1 (bottom level node (51))only. FIG. 2 shows an exemplary table 2 containing historical sales datafor Item I at the bottom level nodes (11) and (12). Similar tables existfor the sale of the other items at their respective locations.

[0053]FIG. 3 shows a computer implemented system 3 with a processor 4,memory 6, a user interface 7 including suitable input devices, forexample, a keypad, a mouse, and the like, and output means, for example,a screen, a printer, and the like, with other computer components forenabling operation of the system including result analysis. The computerimplemented system 3 includes a database 8 for storing historical timeseries of sales information of items at locations, a forecast engine 9for forecasting the mean demand λ_(ij) for each i^(th) consumer item ateach j^(th) location on the basis of the historical sales data, a unitcost file 11 for storing the unit retail price p_(ij), the unitproduction cost c_(ij), the unit return cost g_(ij), and the unitstockout cost b_(ij) of each it consumer item at each j^(th) location ofthe single period inventory system; and an optimization application 12for receiving a threshold for the expected value of a stochastic metricconstraint imposed on a single period inventory system, and fordetermining the optimal constrained draw matrix D{circumflex over ( )}for the most profitable distribution policy for the single periodinventory system subject to the constraint. The computer implementedsystem 3 also includes a Decision Support Tool (DST) 13 for facilitatinguser determination of a distribution policy for a single periodinventory system. The forecast engine 9 may be implemented and managedas illustrated and described in commonly assigned co-pending U.S. patentapplication Ser. No. 10/058,830 entitled “Computer Implemented Methodand System for Demand Forecast Applications”, the contents are which areincorporated herein by reference. Whilst the present invention is beingdescribed in the context of a fully functional computer implementedsystem, it is capable of being distributed as a program product in avariety of forms, and the present invention applies equally regardlessof the particular type of signal bearing media used to carry outdistribution. Examples of such media include recordable type media e.g.CD ROM and transmission type media e.g. digital communication links.

[0054] The present invention will now be exemplified for an exemplary“newsvendor” problem for determining the most profitable distributionpolicy for a single period inventory system having three locations,namely, j=1, 2 and 3 for the sale of newspapers under a stochasticmetric constraint. Assuming that p_(j)=US$0.50, b_(j)=c_(j)=US$0.25, andg_(j)=US$0.00 for all j throughout the single period newspaper inventorysystem, the optimal common availability A* for the most profitabledistribution policy therefor using Eqn. (2) is given by:

A*=(0.5−0.25+0.25)/(0.5+0.25)=0.5/0.75=66.7%.

[0055] Assuming that the demand for newspapers at all locations has aPoisson probability distribution, and the mean demand for a given day isforecasted to be the vector λ=(10, 21, 42), then consequently, theoptimal draw vector D* for the most profitable distribution policy forthat day is D*=(11, 23, 45), namely, a total draw of 79 copies. Usingthe expressions ES(λ_(j),D_(j))=D_(j)−ER(λ_(j),D_(j)) whereER(λ_(j),D_(j))=D_(j)f(λ_(j),D_(j)−1)+(D_(j)−λ_(j))F(λ_(j),D_(j)−2) forthe assumed Poisson distribution of demand, andEST(λ_(j),D_(j))=λ_(j)−ES(λ_(j),D_(j)), the expected sales quantitiesfor the single period newspaper inventory system are 9.2, 20.0 and 40.6at the locations j=1, 2 and 3, respectively, whilst the expected returnsthereat are 1.8, 3.0 and 4.4 respectively, whilst the expected stockoutsthereat are 0.8, 1.0 and 1.4 respectively To summarize, the mostprofitable distribution policy for the single period newspaper inventorysystem leads to an expected total sales quantity of 9.2+20.0+40.6=69.8newspaper copies out of a total draw of 79 newspaper copies, expectedtotal returns of 9.2 newspaper copies, and expected total stockout of3.2 newspaper copies.

[0056] The optimization application 12 can be programmed to determinethe optimal constrained draw vector D{circumflex over ( )} for thesingle period newspaper inventory system to meet a user set sales targetG within a user set predetermined tolerance Δ using either a Lagrangemultiplier approach (see FIG. 4) or an one-by-one allocation ofadditional newspaper copies approach (see FIG. 5). Alternatively, theoptimization application 12 can be programmed to determine the optimalconstrained draw vector D{circumflex over ( )} for the single periodnewspaper inventory system to meet a user set stockout constraint Hwithin a user set predetermined tolerance Δ using either a Lagrangemultiplier approach (see FIG. 6) or an one-by-one allocation ofadditional newspaper copies approach (see FIG. 7). Alternatively, theoptimization application 12 can be programmed to determine the optimalconstrained draw vector D{circumflex over ( )} for the single periodnewspaper inventory system to meet a user set returns constraint Hwithin a user set predetermined tolerance Δ using either a Lagrangemultiplier approach (see FIG. 8) or an one-by-one allocation ofadditional newspaper copies approach (see FIG. 9).

[0057] To determine the optimal constrained draw matrix D{circumflexover ( )} to meet the sales target G of, say, 72 newspaper copies, usingthe Lagrange multiplier approach, on substitution of the relevant costsfrom the unit cost file 9, the optimization application 12 solves thefollowing Lagrange system of equations to meet a sales target G=72 witha predetermined tolerance Δ=0.10:${F\left( {\lambda_{j}D_{j}^{\hat{}}} \right)} = {{\frac{0.5 + M}{0.75 + M}\quad {for}\quad j} = {{1,2\quad {and}\quad 3,{and}\quad {{ES}\left( {\lambda,D^{\hat{}}} \right)}} = 72}}$

[0058] The optimization application 12 uses the binary search method forfinding the next Lagrange multiplier M with the initial value of theLagrange multiplier M set to be the midpoint between the two values of Mobtained by assigning the near extreme values 0.1 and 0.99 to the commonavailability A. To determine the optimal constrained draw matrixD{circumflex over ( )} to meet a sales target G-72 using the one-by-oneadditional allocation approach, on substitution of the relevant costsfrom the unit cost file 11, the optimization application 12 calculatesthe decrease in the expected total profit due to the allocation of anadditional draw unit to each location j=1, 2, and 3 in accordance withthe expression ETP=Σ(0.25D_(j)−0.5ER(λ_(j),D_(j))−0.25EST(λ_(j),D_(j)))whereER(λ_(j),D_(j))=D_(j)f(λ_(j),D^(j)−1)+(D^(j)−λ_(j))F(λ_(j),D_(j)−2) dueto the assumed Poisson distribution of demand andEST(λ_(j),D_(j))=λ_(j)−D_(j)+ER(λ_(j),D_(j)).

[0059] The optimization application 12 takes six iterations to arrive atthe optimal constrained draw vector D{circumflex over ( )}=(14, 26, 50)using the Lagrange multiplier approach (see FIG. 10) and ten iterationsto arrive at the optimal constrained draw vector D{circumflex over( )}=(14, 26, 49) using the one-by-one allocation of additionalnewspaper copies approach (see FIG. 11). In FIG. 11, the winninglocation of each additional draw unit is designated by its incrementalprofit being highlighted in bold. In the present case, the latterapproach requires more steps than the former approach for arriving atits optimal constrained draw vector D{circumflex over ( )} but itachieves the required sales target with one less newspaper copy whichcan be representative of considerable savings for a large single periodnewspaper inventory system delivering, say, two million newspaper copiesdaily. This saving of a newspaper copy is achieved by virtue of theone-by-one allocation repeatedly assigning units to locations to reachthe objective of maximum total profit with minimum draws until theconstraint is satisfied leading to locations having potentiallyconsiderably different availabilities whilst against this, the Lagrangemultplier approach is based on repeatedly calculating common locationavailabilities for determining draws until the constraint is satisfied.

[0060] The optimization application 12 can determine the optimalconstrained draw matrix D{circumflex over ( )} to meet a stockoutconstraint H of, say, 2.5 newspaper copies, or a returns constraint Vof, say, 8.5 newspaper copies in a similar manner as describedhereinabove for meeting the sales target G of 72 newspaper copies.

[0061]FIG. 12 shows the operation of the Decision Support Tool 13 forgraphically plotting the expected total sales ES and the expected totalprofit ETP for a range of total draws centered around the total draw of79 newspaper copies of the most profitable distribution policy for thepresent newspaper single period inventory system. The draw vectors D fordifferent total draw values between 60 and 100 in intervals of 5newspaper copies are preferably arrived at by one-by-one allocation ofnewspaper copies to the optimal draw vector D* for total draws greaterthan 79, and removal of draw units therefrom for total draws less than79. The values of the stochastic metrics ES and ETP at the differenttotal draw values are respectively given by ES(λ,D)=TD-ER(λ,D), andETP(λ,D)=0.25TD−0.50ER(λ,D)−0.25EST(λ,D) on substitution of the abovecosts p_(j)=US$0.50, b_(j)=c_(j)=US$0.25, and g_(j)=US$0.00, from theunit cost file 11, and thereafter the ES and ETP plots are interpolatedbetween the calculated values.

[0062] In addition to and/or instead of plotting the expected totalprofit ETP, the Decision Support Tool 13 can graphically plot theexpected total bookkeeping profit ETBP defined as being equal to theexpected total profit ETP but excluding stockout losses which areopportunity losses as opposed to actual bookkeeping revenues andexpenses. Therefore, in the present single period newspaper inventorysystem, ETBP(λ,D)=0.25TD−0.5ER(λ,D) on substitution of the above costsp_(j)=US$0.50, b_(j)=c_(j)=US$0.25, and g_(j)=US$0.00. As shown, themaximal ETBP value is both higher than the maximal ETP value, and occursat a smaller total draw by virtue of it excluding stockout losses.

[0063] One use of the Decision Support Tool 13 will now be describedwith reference to the following table with values obtained from FIG. 12:TD ES ETP Reduction ETBP Reduction 79 69.8 — — 84 71 0.25 0.45 89 720.85 1.30

[0064] The sales target G of 72 newspaper copies leads to an ETPreduction of about US$0.85 and an ETBP reduction of US$1.30 compared tothe most profitable distribution policy. These reductions may beconsidered as being too great, and therefore a user may use the plots tosettle for a lower sales target of, say, 71 newspaper copies leading tosmaller ETP and ETBP reductions of US$0.25 and US$0.45, respectively,compared to the most profitable distribution policy.

[0065] Another use of the Decision Support Tool 13 is to enable a userto determine a distribution policy for a single period inventory systemsubject to different total draw constraints. For example, the DecisionSupport Tool 13 graphically shows that it is more profitable to delivera total draw of 75 newspaper copies as opposed to 85 newspaper copies,the former and latter constraints leading to ETP reductions of aboutUS$0.14 and US$0.35, respectively.

[0066] While the invention has been described with respect to a limitednumber of embodiments, it will be appreciated that many variations,modifications, and other applications of the invention can be madewithin the scope of the appended claims.

1. Computer implemented system for determining the most profitabledistribution policy for a single period inventory system subject to astochastic metric constraint, the system comprising: (a) a database forstoring historical sales data of sales information of each i^(th)consumer item where i=1, 2, . . . , m at each j^(th) location where j=1,2, . . . , n of the single period inventory system; (b) a forecastengine for forecasting the mean demand λ_(ij) of each i^(th) consumeritem at each j_(th) location of the single period inventory system onthe basis of the historical sales data; (c) a unit cost file for storingthe unit retail price p_(ij), the unit production cost c_(ij), the unitreturn cost g_(ij), and the unit stockout cost b_(ij) of each i^(th)consumer item at each j^(th) location of the single period inventorysystem; and (d) an optimization application for receiving a thresholdfor the stochastic metric constraint imposed on the single periodinventory system, and determining the optimal constrained draw matrixD{circumflex over ( )} for the most profitable distribution policy forthe single period inventory system satisfying the threshold.
 2. Thesystem according to claim 1 wherein the stochastic metric constraint isa sales target ES(λ,D)≧G for some pre-determined threshold G.
 3. Thesystem according to claim 2 wherein, in the case that ES(λ,D*)<G, saidoptimization application numerically solves the Lagrange system ofequations: $\begin{matrix}{{{F\left( {\lambda_{i\quad j},D_{i\quad j}} \right)} = {\frac{p_{i\quad j} - c_{i\quad j} + b_{i\quad j} + {Mw}_{i\quad j}}{p_{i\quad j} - g_{i\quad j} + b_{i\quad j} + {Mw}_{i\quad j}}\quad {for}\quad {all}\quad i}},j} \\{and} \\{{{\sum\limits_{i\quad j}{w_{i\quad j}\left( {D_{i\quad j} - {{ER}\left( {\lambda_{i\quad j},D_{i\quad j}} \right)}} \right)}} = G}\quad}\end{matrix}$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 4. The system according to claim 2 wherein, in the case thatES(λ,D*)<G, said optimization application repeatedly allocatesadditional draw units one-by-one to draws D_(ij) of i^(th) j^(th)item-locations of the single period inventory system for determining theoptimal constrained draw matrix D{circumflex over ( )}, each additionaldraw unit being allocated so as to result in the least decrease in theexpected total profit attributable thereto.
 5. The system according toclaim 4 wherein said optimization application initiates the allocationprocedure from the optimal draw matrix D* for the single periodinventory system.
 6. The system according to claim 1 wherein thestochastic metric constraint is a stockout constraint EST(λ,D{circumflexover ( )})≦H for some predetermined threshold H.
 7. The system accordingto claim 6 wherein, in the case that EST(λ,D*)>H, said optimizationapplication numerically solves the Lagrange system of equations:${{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij} - {Mw}_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j$and${\sum\limits_{ij}^{\quad}{w_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} = H$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 8. The system according to claim 6 wherein, in the case thatEST(λ,D*)>H, said optimization application repeatedly allocatesadditional draw units one-by-one to draws D_(ij) of i^(th) j^(th)item-locations of the single period inventory system for determining theoptimal constrained draw matrix D{circumflex over ( )}, each additionaldraw unit being allocated so as to result in the least decrease in theexpected total profit attributable thereto.
 9. The system according toclaim 8 wherein said optimization application initiates the allocationprocedure from the optimal draw matrix D* for the single periodinventory system.
 10. The system according to claim 1 wherein thestochastic metric constraint is a returns constraint ER(λ,D{circumflexover ( )})≦V for some pre-determined threshold V.
 11. The systemaccording to claim 10 wherein, in the case that ER(λ,D*)>V, saidoptimization application numerically solves the Lagrange system ofequations:$\left. {{{{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j}{and}{\sum\limits_{ij}^{\quad}{w_{ij}{{ER}\left( {\lambda_{ij},D_{ij}} \right)}}}} \right) = V$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 12. The system according to claim 10 wherein, in the case thatER(λ,D*)>V, said optimization application repeatedly removes draw unitsone-by-one from draws D_(ij) of i^(th) j^(th) item-locations of thesingle period inventory system for determining the optimal constraineddraw matrix D{circumflex over ( )}, each removed draw unit beingselected so as to result in the least decrease in the expected totalprofit attributable thereto.
 13. The system according to claim 12wherein said optimization application initiates the removal procedure ofdraw units from the optimal draw matrix D* for the single periodinventory system.
 14. The system according to claim 1 wherein saidoptimization application communicates the expected values of at leastone stochastic metric for at least two draws together with theirassociated expected total profit and/or expected total bookkeepingprofit (ETBP) respectively given by: $\begin{matrix}{{ETP} = {{\sum\limits_{ij}^{\quad}{EP}_{ij}} = {\sum\limits_{ij}^{\quad}{\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}} - {b_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack \quad {and}}}}} \\{{ETBP} = {{\sum\limits_{ij}^{\quad}{EBP}_{ij}} = {\sum\limits_{ij}^{\quad}{\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack.}}}}\end{matrix}$


15. The system according to claim 14 wherein the at least one stochasticmetric is one or more of the following: expected weighted total sales,expected weighted total stockout, and expected weighted total returns.16. The system according to claim 1 wherein the consumer item is aprinted media publication.
 17. Optimization application operating on atleast one computer for determining the most profitable distributionpolicy for a single period inventory system subject to a stochasticmetric constraint wherein: the optimization application is operable toreceive a threshold for the stochastic metric constraint, and theoptimization application is operable to determine the optimalconstrained draw matrix D{circumflex over ( )} for the most profitabledistribution policy for the single period inventory system satisfyingthe threshold.
 18. The optimization application according to claim 17wherein the stochastic metric constraint is a sales targetES(λ,D{circumflex over ( )})≧G for some pre-determined threshold G. 19.The optimization application according to claim 18 and, in the case thatES(λ,D*)<G, operable to numerically solve the Lagrange system ofequations:${{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij} + {Mw}_{ij}}{p_{ij} - g_{ij} + b_{ij} + {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j$and${\sum\limits_{ij}^{\quad}{w_{ij}\left( {D_{ij} - {{ER}\left( {\lambda_{ij},D_{ij}} \right)}} \right)}} = G$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 20. The optimization application according to claim 18 and, in thecase that ES(λ,D*)<G, operable to repeatedly allocate additional drawunits one-by-one to draws D_(ij) of i^(th) j^(th) item-locations of thesingle period inventory system for determining the optimal constraineddraw matrix D{circumflex over ( )}, each additional draw unit beingallocated so as to result in the least decrease in the expected totalprofit attributable thereto.
 21. The optimization application accordingto claim 20 and operable to initiate the allocation procedure from theoptimal draw matrix D* for the single period inventory system.
 22. Theoptimization application according to claim 17 wherein the stochasticmetric constraint is a stockout constraint EST(λ,D{circumflex over( )})≦H for some predetermined threshold H.
 23. The optimizationapplication according to claim 22 and, in the case that EST(λ,D*)>H,operable to numerically solve the Lagrange system of equations:${{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij} - {Mw}_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j$and${\sum\limits_{ij}^{\quad}{w_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} = H$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 24. The optimization application according to claim 22 and, in thecase that EST(λ,D*)>H, operable to repeatedly allocate additional drawunits one-by-one to draws D_(ij) of i^(th) j^(th) item-locations of thesingle period inventory system for determining the optimal constraineddraw matrix D{circumflex over ( )}, each additional draw unit beingallocated so as to result in the least decrease in the expected totalprofit attributable thereto.
 25. The optimization application accordingto claim 23 and operable to initiate the allocation procedure from theoptimal draw matrix D* for the single period inventory system.
 26. Theoptimization application according to claim 17 wherein the stochasticmetric constraint is a returns constraint ER(λD{circumflex over ( )})<Vfor some pre-determined threshold V.
 27. The optimization applicationaccording to claim 26 and, in the case that ER(λ,D*)>V, operable tonumerically solve the Lagrange system of equations:$\left. {{{{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j}{and}{\sum\limits_{ij}^{\quad}{w_{ij}{{ER}\left( {\lambda_{ij},D_{ij}} \right)}}}} \right) = V$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 28. The optimization application according to claim 26 and, in thecase that ER(λ,D*)>V, operable to repeatedly remove draw unitsone-by-one from draws D_(ij) J^(th) item-locations of the single periodinventory system for determining the optimal constrained draw matrixD{circumflex over ( )}, each removed draw unit being selected so as toresult in the least decrease in the expected total profit attributablethereto.
 29. The optimization application according to claim 28 andoperable to iniate the removal procedure of draw units from the optimaldraw matrix D* for the single period inventory system.
 30. Theoptimization application according to claim 17 and operable tocommunicate the expected values of at least one stochastic metric for atleast two draws together with their associated expected total profitand/or expected total bookkeeping profit (ETBP) respectively given by:$\begin{matrix}{{ETP} = {{\sum\limits_{ij}^{\quad}{EP}_{ij}} = {\sum\limits_{ij}^{\quad}{\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}} - {b_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack \quad {and}}}}} \\{{ETBP} = {{\sum\limits_{ij}^{\quad}{EBP}_{ij}} = {\sum\limits_{ij}^{\quad}{\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack.}}}}\end{matrix}$


31. The optimization application according to claim 30 wherein the atleast one stochastic metric is one or more of the following: expectedweighted total sales, expected weighted total stockout, and expectedweighted total returns.
 32. The optimization application according toclaim 17 wherein the consumer item is a printed media publication. 33.Computer implemented method for determining the most profitabledistribution policy for a single period inventory system subject to astochastic metric constraint, the method comprising the steps of: (a)storing historical sales data of the sales information of each i^(th)consumer item where i=1, 2, . . . , m at each i^(th) location where j=1,2, . . . , n of the single period inventory system; (b) forecasting themean demand λ_(ij) of each i^(th) consumer item at each j^(th) locationof the single period inventory system on the basis of the historicalsales data; (c) receiving the unit retail price p_(ij), the unitproduction cost c_(ij), the unit return cost g_(ij), and the unitstockout cost b_(ij) of each i^(th) consumer item at each i^(th)location of the single period inventory system (d) receiving a thresholdfor the stochastic metric constraint; (e) determining the optimalconstrained draw matrix D{circumflex over ( )} for the most profitabledistribution policy for the single period inventory system satisfyingthe threshold; and (f) communicating the optimal constrained draw matrixD{circumflex over ( )} for the most profitable distribution policy forthe single period inventory system.
 34. The method according to claim 33wherein the stochastic metric constraint is a sales targetES(λ,D{circumflex over ( )})≧G for some pre-determined threshold G. 35.The method according to claim 34 wherein, in the case that ES(λ,D*)<G,step (e) includes numerically solving the Lagrange system of equations:${{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij} + {Mw}_{ij}}{p_{ij} - g_{ij} + b_{ij} + {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j$and${\sum\limits_{ij}^{\quad}{w_{ij}\left( {D_{j} - {{ER}\left( {\lambda_{ij},D_{ij}} \right)}} \right)}} = G$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 36. The method according to claim 34 wherein, in the case thatES(λ,D*)<G, step (e) includes repeatedly allocating additional drawunits one-by-one to draws D_(ij) of i^(th) j^(th) item-locations of thesingle period inventory system for determining the optimal constraineddraw matrix D{circumflex over ( )}, each additional draw unit beingallocated so as to result in the least decrease in the expected totalprofit attributable thereto.
 37. The method according to claim 36wherein step (e) includes initiating the allocation procedure of drawunits from the optimal draw matrix D* for the single period inventorysystem.
 38. The method according to claim 33 wherein the stochasticmetric constraint is a stockout constraint EST(λ,D)≦H for somepredetermined threshold H.
 39. The method according to claim 38 whereinin the case that EST(λ,D*)>H, step (e) includes numerically solving theLagrange system of equations:${{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij} - {Mw}_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j$and${\sum\limits_{ij}^{\quad}{w_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} = H$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 40. The method according to claim 38 wherein in the case thatEST(λ,D*)>H, step (e) includes repeatedly allocating additional drawunits one-by-one to draws D_(ij) of i^(th) j^(th) item-locations of thesingle period inventory system for determining the optimal constraineddraw matrix D{circumflex over ( )}, each additional draw unit beingallocated so as to result in the least decrease in the expected totalprofit attributable thereto.
 41. The method according to claim 40wherein step (e) includes initiating the allocation procedure of drawunits from the optimal draw matrix D* for the single period inventorysystem.
 42. The method according to claim 33 wherein the stochasticmetric constraint is a returns constraint ER(λ,D{circumflex over ( )})≦Vfor some pre-determined threshold V.
 43. The method according to claim42 wherein, in the case that ER(λ,D*)>V, step (e) includes numericallysolving the Lagrange system of equations:$\left. {{{{F\left( {\lambda_{ij},D_{ij}} \right)} = {\frac{p_{ij} - c_{ij} + b_{ij}}{p_{ij} - g_{ij} + b_{ij} - {Mw}_{ij}}\quad {for}\quad {all}\quad i}},j}{and}{\sum\limits_{ij}^{\quad}{w_{ij}{{ER}\left( {\lambda_{ij},D_{ij}} \right)}}}} \right) = V$

for determining the optimal constrained draw matrix D{circumflex over( )}.
 44. The method according to claim 42 wherein, in the case thatER(λ,D*)>V, step (e) includes repeatedly removing draw units one-by-onefrom draws D_(ij) of i^(ij) j^(ij) item-locations of the single periodinventory system for determining the optimal constrained draw matrixD{circumflex over ( )}, each removed draw unit being selected so as toresult in the least decrease in the expected total profit attributablethereto.
 45. The method according to claim 44 wherein step (e) initiatesthe removal procedure of draw units from the optimal draw matrix D* forthe single period inventory system.
 46. The method according to claim 33and step (f) includes communicating the expected values of at least onestochastic metric for at least two draws together with their associatedexpected total profit and/or expected total bookkeeping profit (ETBP)respectively given by: $\begin{matrix}\begin{matrix}{{ETP} = {\sum\limits_{ij}{EP}_{ij}}} \\{= {\sum\limits_{ij}\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}} - {b_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack}}\end{matrix} \\{and} \\\begin{matrix}{{ETBP} = {\sum\limits_{ij}{EBP}_{ij}}} \\{= {\sum\limits_{ij}{\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack.}}}\end{matrix}\end{matrix}$


47. The method according to claim 46 wherein the at least one stochasticmetric is one or more of the following: expected weighted total sales,expected weighted total stockout, and expected weighted total returns.48. The method according to claim 33 wherein the consumer item is aprinted media publication.
 49. Computer implemented Decision SupportTool (DST) for facilitating user determination of a distribution policyfor a single period inventory system, wherein: the DST is operable toreceive the unit retail price p_(ij), the unit production cost c_(ij),the unit return cost g_(ij), and the unit stockout cost b_(ij) of eachi^(th) consumer item where i=1, 2, . . . , m at each j^(th) locationwhere j=1, 2, . . . , n of the single period inventory system the DST isoperable to calculate the expected values of at least one stochasticmetric realizable by a draw matrix D for the single period inventorysystem for at least two total draws; and the DST is operable to outputthe expected values of the at least one stochastic metric for the atleast two draws together with their associated expected total profit(ETP) and/or the expected total bookkeeping profit (ETBP) respectivelygiven by: $\begin{matrix}\begin{matrix}{{ETP} = {\sum\limits_{ij}{EP}_{ij}}} \\{= {\sum\limits_{ij}\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}} - {b_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack}}\end{matrix} \\{and} \\\begin{matrix}{{ETBP} = {\sum\limits_{ij}{EBP}_{ij}}} \\{= {\sum\limits_{ij}\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack}}\end{matrix}\end{matrix}$

 where D_(ij) is the delivered quantity of the i^(th) consumer item atthe j^(th) location of the single period inventory system,ER(λ_(ij),D_(ij)) is its expected returns therefrom, andEST(λ_(ij),D_(ij)) is its expected stockout thereat.
 50. The DSTaccording to claim 49 wherein the at least one stochastic metric is oneor more of the following: expected weighted total sales, expectedweighted total stockout, and expected weighted total returns.
 51. TheDST according to claim 49 wherein the consumer item is a printed mediapublication.
 52. Computer implemented method for facilitating userdetermination of a distribution policy for a single period inventorysystem, the method comprising the steps of: (a) receiving the unitretail price p_(ij), the unit production cost c_(ij), the unit returncost g_(ij), and the unit stockout cost b_(ij) of each i^(th) consumeritem at each j^(th) location where j=1, 2, . . . , n of the singleperiod inventory system; (b) calculating the expected values of at leastone stochastic metric realizable by a draw matrix D for the singleperiod inventory system for at least two total draws; and (c) outputtingthe expected values of the at least one stochastic metric for the atleast two draws together with their associated expected total profit(ETP) and/or the expected total bookkeeping profit (ETBP) respectivelygiven by: $\begin{matrix}\begin{matrix}{{ETP} = {\sum\limits_{ij}{EP}_{ij}}} \\{= {\sum\limits_{ij}\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}} - {b_{ij}{{EST}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack}}\end{matrix} \\{and} \\\begin{matrix}{{ETBP} = {\sum\limits_{ij}{EBP}_{ij}}} \\{= {\sum\limits_{ij}\left\lbrack {{\left( {p_{ij} - c_{ij}} \right)D_{ij}} - {\left( {p_{ij} - g_{ij}} \right){{ER}\left( {\lambda_{ij},D_{ij}} \right)}}} \right\rbrack}}\end{matrix}\end{matrix}$

 where D_(ij) is the delivered quantity of the i^(th) consumer item atthe j^(th) location, ER(λ_(ij),D_(ij)) is its expected returnstherefrom, and EST(λ_(ij),D_(ij)) is its expected stockout thereat. 53.The method according to claim 52 wherein the at least one stochasticmetric is one or more of the following: expected weighted total sales,expected weighted total stockout, and expected weighted total returns.54. The method according to claim 52 wherein the consumer item is aprinted media publication.